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Math Help - help!!!- optimization problem

  1. #1
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    help!!!- optimization problem

    I dont know how to start on this probem. It would nice if someone helps me.

    A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the developments. One is 3 miles south of the existing line; the other is 4 miles south of the existing line and 5 miles east of the first development. Find the place on the existing line, relative to the 2 developments, to make the connection that minimizes the total length of the new line.

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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by 52090 View Post
    I dont know how to start on this probem. It would nice if someone helps me.

    A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the developments. One is 3 miles south of the existing line; the other is 4 miles south of the existing line and 5 miles east of the first development. Find the place on the existing line, relative to the 2 developments, to make the connection that minimizes the total length of the new line.
    By Pythagoras' theorem, the length of the line connecting the town 3 miles south of the existing line is \sqrt{x^2 + 3^2} and the length of the line connecting the town 4 miles south of the existing line is \sqrt{4^2 + (5 - x)^2}. you want to minimize the sum of those

    hint: the square roots aren't needed, hope you know why
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  3. #3
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    Hello, 52090!

    If we must use Calculus, the approach and set-up is quite ugly.


    A water line runs east-west. A town wants to connect two new housing developments
    to the line by running lines from a single point on the existing line to the developments.
    One is 3 miles north of the existing line; the other is 4 miles north of the existing line
    and 5 miles east of the first development.

    Find the place on the existing line, relative to the two developments,
    to make the connection that minimizes the total length of the new line.
    Code:
                              * B
                            * |
        A *               *   |
          | *           *     | 4
        3 |   *       *       |
          |     *   *         |
          * - - - * - - - - - *
          C   x   P    5-x    D

    Development A is 3 miles from the line: . AC = 3
    Development B is 4 miles from the line: . BD = 4
    CD = 5

    Let P be a point on CD.
    Let x \,=\, CP, then 5-x \,=\, PD

    We want to minimize the distance: AP + PB

    In right triangle ACP\!:\;AP \:=\:\sqrt{x^2+3^2}
    In right triangle BDP\!:\;PB \:=\:\sqrt{(5-x)^2 + 4^2}

    The total distance is: . D(x) \;=\;\left(x^2+9\right)^{\frac{1}{2}} + \left(x^2-10x+41\right)^{\frac{1}{2}}

    . . and that is the function we must minimize.



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